Multi target tracking initiation with passive angle measurements

ABSTRACT

The present invention is in general related to tracking of multiple targets by means of measurements by various sensors. In particular the invention provides procedures for track initiation during multiple target tracking by means of measurements from passive sensors. The invention defines a quality measure for each tentative new target, by which the tentative targets are sorted and selected. The calculation of the parameters of possible targets and their covariance are preferably performed in a recursive manner. The track initiation comprises the steps of creating strobe tracks, calculating strobe track crosses, selecting a strobe track cross as a probable target and creating a target track.

TECHNICAL FIELD OF THE INVENTION

The present invention relates in general to tracking of multiple targetsby means of measurements from various sensors and in particular to trackinitiation during multiple target tracking by means of measurements frompassive sensors.

PRIOR ART

Traditionally, tracking has been performed using measurements fromactive sensors, such as radar's or active sonar's, which reportmeasurements from different sources. These sources may be targets ofinterest as well as noise or false targets. Tracking serves the objectto organise the sensor data into sets of observations, hereinafterdenoted tracks, produced by the same source. Once the existence of atrack has been established it is possible to estimate relatedquantities, such as the target position, velocity, acceleration as wellas other specific characteristics.

The basics of multiple target tracking include three phases; trackinitiation, track maintenance and track deletion. Track initiationinvolves processes in which a set of single measurements are collectedand a likelihood that they originate from the same source is calculated.When such a probability is high enough, a track is created andassociated with a probable target. Track maintenance comprisescalculations of track or target characteristics, but may also be used topredict the behaviour of the target in a near future. Such estimates areoften computed by filtering of a series of similar measurements over acertain time period, since the individual measurements often containmeasurement errors and noise. These calculations normally involveprevious measurements, condensed into the so called states of the track,or predictions as well as new measurements from the sensors. This meansthat once a track is created, it “consumes” new measurements, which fallclose enough to the predicted characteristics of the target, and suchmeasurements are not used to initiate new tracks.

Even if a target disappears, or at least avoids being detected, thetrack will survive for a certain time, in order to handle misseddetection's or shorter malfunctions. However, the estimates of the trackcharacteristics deteriorate, and so do the predictions. When theestimates and predictions become too uncertain, the track is no longerof use, and should be deleted. Such track deletion may be based oncalculated uncertainty levels of the estimated track parameters or on acertain numbers of “missing” observations.

The tracking step incorporates the relevant measurements into theupdated track parameter estimates. Predictions are made to the time whenthe next data set is to be received. This prediction constitutes theorigin from which the determination of whether a new measurement fitsinto the track or not is made. The selection of new measurements asbelonging to the track or not is known as “gating” or measurementassociation. The prediction typically constitutes the middle of thegate, and if the measurement falls within a certain gate width, it willfor example be assumed to belong to the track. A common way to performestimation and prediction is by employing Kalman filtering. Furtherreferences to Kalman filtering can be found in “Estimation and Tracking:Principles, Techniques, and Software” by Bar-Shalom and Li, ArtechHouse, USA, 1993, page 209 to 221.

A description of tracking systems of prior art can e.g. be found in“Multiple-Target Tracking with Radar Applications” by Samuel S.Blackman, Artech House, USA, 1986, page 4 to 11.

In prior art multi target systems, radar is often used. Radarmeasurements provide information about azimuth angle and range (2Dradar's), and in most cases even the elevation (3D radar's), withrespect of the sensor position. It will be understood that from suchmeasurements, estimates of target positions, velocities etc. are easilyobtainable within the above described scheme.

In modern tracking systems, especially in military applications, the useof radar measurements is not solely of benefit. Since the radar is anactive sensor, it radiates energy and records reflected waves, fromwhich the position can be determined. However, such radiating sourcesare easily located by enemies and may therefore be destroyed by missilesor assist in the navigation of an hostile target. It is thereforeadvantageous if tracking would be possible to perform using only passivesensors, such as jam strobes from the targets, ESM (Electronic SupportMeasures) sensors or IR-/EO-sensors (infrared/ElectroOptical). A majordisadvantage with the passive sensors as compared with radar is thatthey do not have any possibility to detect any range information from asingle sensor. They will normally only provide measurements of theazimuth (1D sensor) or azimuth and elevation (2D sensor), with respectof the sensor location.

An obvious approach to overcome such a problem is to employ at least twosensors, separated by a distance, and use the combination of themeasurements. By this it is possible to perform a geometricaltriangulation, which at least in principle may give the absolutepositions of the target as the intersection point between twomeasurement directions. The measurement directions are hereinafterreferred to as “strobes”, and the intersections are denoted as“crosses”. However, if there are several targets present in the area atthe same time, pure geometrical considerations are not enough to findthe unique target positions, since there generally are more crossesbetween strobes than true targets. A cross that does not correspond toany true target is denoted a “ghost”. Furthermore, since themeasurements are corrupted by errors, strobes including both azimuth andelevation may not even intersect each other exactly. Thus, there is aneed for a process in which the true targets among the crosses areidentified and in which the ghosts are rejected.

A possible way to solve this problem is to calculate all possiblecrosses from all possible strobes and formulate a maximum likelihoodproblem. Such a problem may be solved in a conventional way bycomputers, but using a number of sensors tracking a number of targetswill produce very large number of crosses. The computer time which isneeded for such calculations will grow tremendously with the number oftargets and the number of sensors, and even for relatively moderatenumbers, the calculations will be impossible to perform on computers oftoday in real time. It is obvious for someone skilled in the art that atracking system that cannot perform in real time is of no use.

In the U.S. Pat. No. 4,806,936 a method of determining the positions ofmultiple targets using bearing-only sensors only is disclosed. In thismethod, individual strobe measurements from three sensors are used. Theintersecting bearing lines form triangles representing both true targetsand ghosts. The separation of the ghosts from the true targets isperformed by analysing the size and position of each triangle and ingating processes eliminate some of the ghosts. The remaining set oftriangles is entered into a maximum likelihood procedure to extract thetrue targets. The gating process is based on simple geometricalmeasures, such as the difference between the individual strobes and thegeometrical centre of gravity of the triangles. Such measures arehowever sensitive to measurement uncertainties since an uncertainmeasurement will enter the calculations with the same computationalweight as the more accurate ones. Since the individual strobes, whichnormally involve large measurement uncertainties, are used for thesecalculations, the determinations of position of the true targets can notbe performed very accurately. Furthermore, the assumption that theremust exist a detection from three individual sensors will limit therange of detection significantly. It is also not obvious how to make ageneralisation to more than three sensors. An obvious disadvantage withthe above method is also that all the sensors have to be synchronised inorder to allow a comparison between the individual strobes. Sensorsworking at different rates or with different offset times can not beused together with the above described method without introducing largeerrors.

SUMMARY OF THE INVENTION

An object of the present invention is to provide a method formulti-target tracking using passive sensor measurements from at leasttwo sensors for track initiation, which method is possible to perform inreal time and does not exhibit the above disadvantages.

The object of the present invention is achieved by a process exhibitingthe features set forth in the claims. The process of the invention usesfiltered strobe tracks, which give accurate angle determinations as wellas angle velocities, accelerations and other relevant quantities, as thesource for creating strobe track crosses. The invention preferablydefines a quality measure for each strobe track cross, representing atentative new target, by which quality measure the tentative targets aresorted and selected. The quality measure is based on the consistency ofstrobe track parameters, such as angles and angular velocities. Thecalculation of the positions, speeds and related quantities of possiblestrobe track crosses and the covariance of their parameters arepreferably performed in a recursive manner.

DRAWINGS

Embodiments according to the present invention are presented in detailin the following, in connection with the associated drawing, in which:

FIG. 1 is a block diagram illustrating the multi-target trackingprocess;

FIG. 2 is a block diagram illustrating the target track initiationprocess;

FIG. 3 is a block diagram illustrating the strobe-tracking process;

FIG. 4 is a schematic illustration of strobes and strobe tracks;

FIG. 5 is a schematic illustration of the recursive manner in whichhigher order strobe track crosses are calculated;

FIG. 6 is a schematic illustrating of strobe track crosses with 1Dsensors;

FIG. 7 is a schematic illustrating of strobe track crosses with 2Dsensors;

FIG. 8 is a block diagram illustrating the selection of strobe trackcrosses as probable targets;

FIG. 9 is a block diagram illustrating the initiation process;

FIG. 10 is diagrammatic illustration of a tracking situation with twosensors and one true target;

FIG. 11 is diagrammatic illustration of a tracking situation with threesensors and one true target;

FIG. 12 is diagrammatic illustration of a tracking situation with threesensors and two true targets;

FIG. 13 is diagrammatic illustration of a part of the tracking situationin FIG. 12; and

FIG. 14 illustrates the sensor and ET coordinate system;

DETAILED DESCRIPTION

For two vectors u=(u₁, u₂, u₃) and v=(v₁, v₂, v₃) in ³ the scalar, crossand tensor product are defined as follows

u·v=u₁v₁+u₂v₂+u₃v₃,

u×v=(u₂v₃−u₃v₂, u₃v₁−u₁v₂, u₁v₂−u₂v₁) and

${u \otimes v} = {\begin{pmatrix}{u_{1}v_{1}} & {u_{1}v_{2}} & {u_{1}v_{3}} \\{u_{2}v_{1}} & {u_{2}v_{2}} & {u_{2}v_{3}} \\{u_{3}v_{1}} & {u_{3}v_{2}} & {u_{3}v_{3}}\end{pmatrix}.}$

In FIG. 14 the used coordinate systems are illustrated. The ET system isa global coordinate system where the system tracks are defined andstrobe tracks from different sensors are compared and strobe trackcrosses are computed. Strobes and strobe tracks has a simple canonicalrepresentation in the appropriate sensor coordinate system.

The ET system is normally used with Cartesian coordinates. Each sensoris associated with a sensor coordinate system (SS), where the foot pointof the sensor is positioned at the origin of the SS. The position of thefoot point in the ET system is represented by a vector F. The strobetracks in the SS has a natural expression in polar coordinates. Such apolar point has the following representation in the Cartesian system:

(r,θ,φ)→(r sin(θ)cos(φ),r cos(θ)cos(φ),r sin(φ)).

θ represents the azimuth and ranges from 0 to 2π, φ represents theelevation angle and ranges from −π/2 to π/2, where 0 corresponds to thehorizontal plane, and finally r represents the range (from 0 toinfinity). The linear transformation from ET to SS is given by

TX=AX+F,  (1)

i.e a rotation A plus a translation F.

With reference to FIG. 1, the multi-target tracking process for passivesensors follows the same basic steps as in prior art tracking processesfor active sensors. The process involves the steps of target trackinitiation 1, target track maintenance 2 and target track deletion 3.

In FIG. 2, the main steps of the process of target track initiationaccording to the present invention are illustrated. The four main stepscomprise creating strobe tracks 11, calculating strobe track crosses 12,selecting strobe track crosses 13 as probable targets, and finallycreating a target track 14. The initiation process therefore starts witha bunch of individual strobe measurements and ends with the creation ofa target track.

A strobe is defined as an individual measurement from a single sensorand comprises basically the angle to a signal source and ischaracterised by certain measurement time. If the sensor is of a 1Dtype, only the azimuth angle is available, while in the case of a 2Dsensor, both the azimuth angle and the elevation angle are measured.

A strobe track is a filtered set of strobes, which belong to the sametarget. A strobe track is associated with a strobe track state, whichcomprises estimates of angle, angular velocity and other relevantparameters of the target as well as their covariances, based on theindividual strobes associated with the same target. From the strobetrack state, predictions to a near future is also possible to perform,assuming a certain dynamical model.

The initial step of creating strobe tracks 11 is shown in detail in FIG.3, and follows basically the same pattern as the full tracking processusing active sensors. The strobe track creation step 11 thus involvesstrobe track initiation 21 for each target, associating incoming strobeswith appropriate strobe tracks 22, updating and predicting strobe tracks23 and strobe track deletion 24. The strobe tracking process differsfrom tracking processes for active sensors in that only estimates ofangular parameters are available, and not full position information.However, angles, angular velocities and, if applicable, angularaccelerations are estimated within the strobe track state.

The strobe tracking process takes place for each single sensorindependent of the other sensors. The whole process starts with a seriesof measurements from the sensor. At certain azimuth angles, and ifapplicable at certain elevations, detection's of a tentative target aremade. The single measurements, i.e. the individual strobes, give certainvalues of the azimuth angle θ and in applicable cases also of theelevation angle φ. Such measurements or observations are collected andwhen a set of measurements are consistent with each other, regardingestimated azimuth angles, azimuth angle velocities, and if applicablealso elevation angle and elevation angle velocity, a strobe track iscreated.

When a strobe track is created, a strobe track state is defined. Thestrobe track state comprises estimates of angle, angular velocity andother parameters which are connected to the target, as well as theircovariances. New incoming strobes are compared to the strobe track stateand if the incoming strobe is consistent with the predicted parametersof the strobe track propagated to that particular measurement time, theincoming strobe is associated with the strobe track. The associationprocess follows the methods known in the art. The incoming strobe isthen used for updating the strobe tracks and for predicting the strobetrack state for future measurement times. Such updating and predictingis preferably performed by filtering, which provides angle, angularvelocity as well as associated covariances.

A particularly preferable way to perform the strobe track maintenance isby Kalman filtering, in which a series of estimates of strobe tracksparameters are created. Estimated values of θ and φ as well as {dot over(θ)}, {dot over (φ)}, {umlaut over (θ)} and {umlaut over (φ)} may becalculated, all referring to the local spherical coordinate system ofthe sensor. The Kalman filtering also has the advantage of providing thevariance of the different estimates and covariance's between angles andangular velocities, thus giving an uncertainty measure of the strobetrack. Once a strobe track is initiated, conventional methods ofmaintaining and deleting tracks are applicable. In this way, one sensormay give rise to several strobe tracks.

As an example, the following model may be used to filter azimuth orelevation in a strobe track. The state vector and its covariance isdescribed as ${\begin{pmatrix}\theta \\\overset{.}{\theta}\end{pmatrix}\quad {and}\quad \begin{pmatrix}P_{\theta\theta} & P_{\theta \overset{.}{\theta}} \\P_{\overset{.}{\theta}\theta} & P_{\overset{.}{\theta}\overset{.}{\theta}}\end{pmatrix}},$

respectively.

The dynamic are simple and at propagation, process noise is addedaccording to {umlaut over (θ)}=white noise. In discrete time for a 1Dscan it is represented as

{dot over (θ)}(t_(i+1))={dot over (θ)}(t_(i))+w_({dot over (θ)})(t_(i))

θ(t_(i+1))=(t_(i+1)−t_(i)){dot over (θ)}(t_(i))+θ(t_(i))+w_(θ)(t_(i))

where w_({dot over (θ)})(t_(i)) and w_(θ)(t_(i)) are noise correspondingto the azimuth angle and angular velocity, respectively, which model theuncertainty of the constant angular velocity motion. The covariancematrix of the noise can be for example be given by (see “Estimation andTracking: Principles, Techniques, and Software” by Bar-Shalom and Li,Artech House, USA, 1993, page 263): $R = {\begin{pmatrix}{\frac{1}{3}( {t_{i + 1} - t_{i}} )^{3}} & {\frac{1}{2}( {t_{i + 1} - t_{i}} )^{2}} \\{\frac{1}{2}( {t_{i + 1} - t_{i}} )^{2}} & ( {t_{i + 1} - t_{i}} )\end{pmatrix}\sigma_{v}^{2}}$

where σv² is a parameter related to the uncertainty of the model. Thefilter update is straight forward using the Kalman filter approach. Themeasurement of the azimuth angle can for example be modelled as

z(t_(i))=θ(t_(i))+e(t_(i))

where z(t_(i)) is the measure at time t_(i) and where e(t_(i)) is themeasurement distribution with variance λ². This quantity can be obtainedfrom sensor characteristics.

The above description fully defines the Kalman filter, as is obvious toanyone skilled in the art.

To formalise, for a sensor that yields 2D-strobes (θ,φ) (azimuth andelevation) as measurements, a 2D-strobe track is an estimator thatalways yield an estimate of the following quantities at any time:$\begin{matrix}{{\begin{pmatrix}\theta \\\overset{.}{\theta}\end{pmatrix}\quad,\quad \begin{pmatrix}P_{\theta\theta} & P_{\theta \overset{.}{\theta}} \\P_{\overset{.}{\theta}\theta} & P_{\overset{.}{\theta}\overset{.}{\theta}}\end{pmatrix}}{\begin{pmatrix}\phi \\\overset{.}{\phi}\end{pmatrix}\quad,\quad \begin{pmatrix}P_{\phi\phi} & P_{\phi \overset{.}{\phi}} \\P_{\overset{.}{\phi}\phi} & P_{\overset{.}{\phi}\overset{.}{\phi}}\end{pmatrix}}} & (2)\end{matrix}$

For a sensor that yields 1D-strobes (θ) (azimuth) as measurements, a1D-strobe track is an estimator that always yield an estimate of thefollowing quantities at any time: $\begin{matrix}{\begin{pmatrix}\theta \\\overset{.}{\theta}\end{pmatrix}\quad,\quad \begin{pmatrix}P_{\theta\theta} & P_{\theta \overset{.}{\theta}} \\P_{\overset{.}{\theta}\theta} & P_{\overset{.}{\theta}\overset{.}{\theta}}\end{pmatrix}} & (3)\end{matrix}$

The above described process of strobe track creation is performed forevery individual sensor. Since the sensors may be of different types,having different scan times, the strobe track creation for eachindividual sensor is performed independent of every other sensor. Thatis, strobe tracks from different sensors may have different updatingrates or offset times.

If no new strobes are associated with an existing strobe track, theuncertainty of the predictions deteriorate with time, and the strobetrack will eventually be deleted. This may happen when a target leavesthe measurement range of a sensor, disappears in any other way, or ifthe strobes are used in any other way, as described later. The strobetrack deletion follows common procedures, and is known per se.

If there are N sensors, we denote the different sensors as D1, D2, . . ., DN. If each sensor gives rise to m1, m2, . . . , mN strobe tracks,these strobe tracks are denoted ST11, ST12, . . . , ST1m1, and ST21,ST22, . . . , ST2m2 and so on.

When a strobe track is formed, at least fairly good estimates of most ofits properties are available, as well as their development in time.Since the sensors may operate with different time intervals, all strobetrack have to be synchronised in order to combine information fromseveral sensors. This is done by a central unit, collecting strobe trackinformation from the different sensors, and propagating the estimatedstrobe track parameters to a common time, i.e. predicting all strobetrack parameters to one specified time. This specified time is normallychosen to be equal to the last of the strobe track updating times.

These predicted strobe track parameters constitute the foundation onwhich the strobe track crosses are created and on which target trackinitiation is based. Note that also the covariance information istransformed in this manner. The information constituted by the strobetracks propagated to a common time is normally more accurate compared toinformation from individual strobes, since they are filtered foraveraging out statistical noise. It is also obvious that informationabout angular velocity, which is not available from individual strobes,is of benefit for the following analysing procedures. Last, but notleast, the covariance of the strobe track parameters are of use inestimating the accuracy in the measurements, and not only statisticaluncertainties, such in the case of individual strobes.

In FIG. 4 two strobe tracks ST11 and ST21 are shown as thick lines,associated with sensors D1 and D2, respectively. The uncertainty of theazimuth angle in each respective coordinate system C1 and C2 is picturedas a probability distribution function PDF11 and PDF21, respectively, atthe end of each strobe track. In the same figure, a number of individualstrobes associated with the strobe tracks S11 (t_(1i)), S21 (t_(2i)) aredrawn with thin lines, indicating the variation of the individualmeasurements. The uncertainty U11 (t_(1i)), U21 (t_(2i)) of each strobeis indicated superimposed with PDF11 and PDF21. In order to simplify thefigure, some of the notations are omitted. At the intersection pointbetween the strobe tracks ST11 and ST12 a strobe track cross X11 isformed. From this picture, the advantage of using strobe tracks insteadof individual strobes for creating crosses is obvious. The overalluncertainty is lower, estimates of angle velocities V11 and V12 of thestrobe tracks are available, and the PDF:s are well established from thefiltering process. Already at this point the strobe track cross X11 maybe given a certain velocity VX11 in the global coordinate system.

The next step in the total process is thus to find the locations of thedifferent strobe track crosses and calculate estimates of the position,velocity and other important properties of strobe track crosses. Astrobe track cross, as above defined, is an intersection point or apoint in the vicinity of closely positioned strobe tracks, which maycorrespond to the most probable position of a true target. If there areonly two sensors, the strobe track crosses can only consist of anintersection or a position near the minimum distance between two strobetracks, one strobe track from each sensor. If there are more sensors,there still exist strobe track crosses between two strobe tracks, butthere will also be intersections or minimum distances between threestrobe tracks or more. The order of a strobe track cross denotes thenumber of strobe tracks that are involved in creating the strobe trackcross, i.e. a strobe track cross formed by two strobe tracks is denoteda 2nd order strobe track cross, a strobe track cross formed by threestrobe tracks is denoted a 3rd order strobe track cross, and so on. Astrobe track cross can be formed from at most one strobe track from eachsensor. Some of these strobe track crosses represent true targets, butmost of them—the ghosts—are just coincidences and do not correspond toany true target. One important object of the present invention is toprovide a reliable way to eliminate these ghosts.

A possible way for finding the strobe track crosses is to use a purecombinatory approach and calculate all geometrical possibilities. Thiswill for a number of targets and a number of sensors easily give a hugenumber of possible combinations, in fact, the problem is NP-hard, whichmeans that the computational complexity grows faster than a polynomialfunction of the number of targets. A preferred approach according to theinvention is instead to use a recursive scheme for calculating thestrobe track crosses and their quality. With reference to FIG. 5,illustrating a case where three sensors are used, the recursive schemestarts with the calculations of the hypothetical 2nd order strobe trackcrosses. These calculations make use of the strobe track states fromeach sensor propagated to a common time, represented by data from sensor1, sensor 2 and sensor 3, respectively. The 2nd order strobe trackcrosses are calculated in a manner described below, if applicableincluding a coarse gating procedure, and stored in the data list ofstrobe track crosses 1&2, strobe track crosses 1&3 and strobe trackcrosses 2&3, respectively. In the next step, the information gatheredfor the hypothetical 2nd order strobe track crosses is used for thecalculation of hypothetical 3rd order strobe track crosses. This meansthat the data of 2-cross 1&2 and the data of 2-cross 1&3 are used tocalculate hypothetical 3rd order strobe track crosses. These parametersare stored in data list of strobe track crosses 1&2&3. In this way,previous calculations are used to reduce the necessary processing powerfor performing the calculation of strobe track crosses.

The generalisation to calculation of higher order strobe track crossesis obvious for someone skilled in the art.

A strobe track in the two-dimensional plane may be visualised as acentral line representing the estimated azimuth angle, and tails on eachside of the central line, representing the decreasing probabilitydensity function values.

A 1D strobe track in the three-dimensional space, may be visualised as acentral vertical plane with descending probability tails on each side.The upper and lower boundaries of the plane is only set by externalconsiderations, such as e.g. minimum and maximum possible flightheights. The mathematical representation of a strobe track using set ofvariables (3) above is as follows: An orthonormal set of cylindercoordinates (e_(ρ), e_(θ), e_(h)) defined as follows in the sensorsystem (SS)

e_(ρ)=(sin(θ), cos(θ), 0),

e_(θ)=(cos(θ), −sin(θ), 0),  (4)

e_(h)=(0, 0, 1).

Now the strobe track can parametrised as follows:

l(d, h)=e_(ρ)d+e_(h)h+F, d>0, h,  (5)

where F is the sensor foot point, i.e. the origin point of the sensorsystem (SS) in the ET system. The 1-dimensional distribution of thestrobe track can be linearised at a point (d₀, h₀) and represented as adegenerate Gaussian distribution in ³:

N(e_(ρ)d₀+e_(h)h₀+F, P),  (6)

where the covariance and its inverse is

P=d₀ ²P_(θθ)e_(θ){circle around (X)}e_(θ),  (7a)

P⁻¹=P_(θθ) ⁻¹e_(θ){circle around (X)}e_(θ)d₀ ⁻²  (7b)

and the linearised distribution function is $\begin{matrix}{{f(x)} = {\frac{1}{\sqrt{2\pi \quad P_{\theta\theta}}}{{\exp ( {{- ( {x - F} )}{{P^{- 1}( {x - F} )}/2}} )}.}}} & (8)\end{matrix}$

A 2D-strobe track in three-dimensional space may be seen as a centralline specified by the estimated azimuth angle and the estimatedelevation angle, which line is surrounded by a cone of the descendingprobability density function. The mathematical representation of a2D-strobe track using definition (1) above is as follows: An orthonormalset of polar coordinates (e_(r), e_(θ), e_(φ)) defined as follows in thesensor system (SS)

e_(r)=(sin(θ)cos(φ), cos(θ)cos(φ), sin(φ)),

 e_(θ)=(cos(θ)cos(φ), −sin(θ)cos(φ), sin(φ)),  (9)

e_(φ)=(−sin(θ)sin(φ), −cos(θ)sin(φ), cos(φ)).

Now the strobe track can parametrised as follows

l(d)=e_(r)d+F, d>0,  (10)

where F is the sensor foot point. The 2-dimensional distribution of thestrobe track can be linearised at a point (d₀) and represented as adegenerate Gaussian distribution in ³:

N(e_(r)d₀+F, P),  (11)

where the covariance and its inverse is

P=(P_(θθ)cos²(φ)e_(θ){circle around (X)}e_(θ)+P_(φφ)e_(φ){circle around(X)}e_(φ))d₀ ².  (12a)

P⁻¹=(P_(θθ) ⁻¹ cos⁻²(φ)e_(θ){circle around (X)}e_(θ,i)+P_(φφ) ⁻¹e_(φ){circle around (X)}e_(φ))d₀ ⁻²  (12b)

and the linearised distribution function is $\begin{matrix}{{f(x)} = {\frac{1}{2\pi \sqrt{P_{\theta\theta}P_{\phi\phi}}}{{\exp ( {{- ( {x - F} )}{{P^{- 1}( {x - F} )}/2}} )}.}}} & (13)\end{matrix}$

For a representation in the ET-system (FIG. 15), the foot point F andthe orthonormal system is computed in the ET-system (c.f. thetransformation relation (1)).

If at least one sensor is a 1D sensor, a true intersection will alwaysexist between two strobe tracks, which are not parallel or diverging. Ifone sensor is a 1D sensor and the other one a 2D sensor, a uniqueintersection point is defined, since it corresponds to an intersectionbetween a line and a half plane in space. If both sensors are 1Dsensors, the intersection will be a line, and additional informationsuch as an assumed height must be added to achieve the full parameterset. Such a situation is diagrammed in FIG. 6. The sensors D1 and D2,working in their respective coordinate systems C1 and C2, have 1D strobetracks ST11 and ST12, respectively, for a certain target T. The onlyavailable angle information is the azimuth angle, and the heightrestrictions are set by external reasons. The strobe track cross X11will thus be a line.

If both sensors are 2D sensors, a true strobe track cross will notalways exist between the lines corresponding to the estimated angles ofthe strobe tracks. Since the strobe track state contain uncertaintiesand noise, it is likely that the strobe tracks just will pass close toeach other, but not intersect each other perfectly. Such a case issketched in FIG. 7. Three strobe tracks ST11, ST21 and ST22 from two 2Dsensors D1 and D2, respectively, are shown. The three cones representsthe areas within which the probability density functions have valueslarger than a certain threshold. The strobe tracks ST11 and ST21 are notso close together that the cones intersect, and it is likely that theassociated strobe track cross (X11) is not a true target. The conesrepresenting the strobe tracks ST11 and ST22 do intersect and a mostprobable position selected as the strobe track cross X12.

The notation of the strobe track crosses follows the below scheme. Theindices correspond to the respective sensor so that the first indexcorresponds to sensor number one, the second index to sensor number 2and so on. For a system with four sensors, there will thus be fourindices. The number denotes the number of the strobe track that is usedto create the strobe track cross. A “0” tells that the particular sensordoes not contribute to that specific strobe track cross. A 2nd orderstrobe track cross thus has two non-zero indices, a 3rd order strobetrack cross has three non-zero indices and so on. In the example of FIG.7, X12 means that the strobe track cross is created by the first strobetrack ST11 from sensor number one D1 and the second strobe track ST22from sensor number two D2.

In order to calculate 2nd order strobe track crosses from the strobetrack states, the following procedure is preferred. First a distancebetween the two strobe tracks is calculated for performing a firstgating. The criteria can be of any kind of distance measure, butpreferably the statistical distance in a Cartesian global coordinatesystem is used. This preferred distance measure, between two 2D-strobetracks, is described below.

The computation of the statistical distance between two 2D-strobe tracksis as follows. The strobe tracks are given by two half lines with thefollowing representation (c.f. relation (10)):

l_(i)(d)=e_(r,i)d+F_(i), d>0, i=1,2.  (14)

The shortest geometrical distance between the strobe tracks is given by$\begin{matrix}{{d^{2}( {l_{1},l_{2}} )} = {{\inf\limits_{d_{1},{d_{2} > 0}}( {{e_{r,1}d_{1}} + F_{1} - {e_{r,2}d_{2}} - F_{2}} )}^{2}.}} & (15)\end{matrix}$

The d₁,d₂ that yield the infimum above gives the point on the strobetrack that is closest to the other strobe track. Their numerical valueis the distance of that point to the sensor position.

Given fixed d₁,d₂ that yields the min distance above, the linearisedstatistical distance between the strobe tracks can be computed as$\begin{matrix}{{StatDist} = \frac{\Delta}{\sqrt{P}}} & (16)\end{matrix}$

where

Δ=(F₁−F₂)·e₀,

P=(P_(θθ,1) (e₀·e_(θ,1))²+P_(φφ,1) (e₀·e_(φ,1))²)d₁²+(P_(θθ,2)(e₀·e_(θ,2))²+P_(φφ,2) (e₀·e_(φ,2))²)d₂ ²

e₀=e_(r,1)×e_(r,2).

The calculated statistical distance value is compared to a predeterminedthreshold value, and if the calculated distance value exceeds thethreshold, the pair of strobe tracks is very unlikely as a candidate fora strobe track cross and is rejected. The predetermined value could be afixed value, a value chosen by the operator, a value depending on theuncertainties of the strobe track states or a combination thereof.

If the pair of strobe tracks pass this first gating process, a closestpoint on respective strobe track, which corresponds to the calculatedminimum distance, is chosen. From these point positions, apoint-to-sensor distance between each closest point and each respectivesensor is calculated. These point-to-sensor distances, d₁ and d₂,respectively, are compared to the expected range of each respectivesensor, and if at least one of these point-to-sensor distances exceedsthe range of the respective sensor, the pair of strobe tracks is assumedto be associated with a ghost cross and is subsequently rejected.

The remaining pairs of strobe tracks, which have passed the two coarsegating processes described above, will give rise to a 2nd order strobetrack cross. The strobe track cross position and the associateduncertainty is calculated. This calculation uses the information whichis available from the strobe track states. Since not only the estimatedangles are available, but also estimated angular velocities andcovariance's associated with these parameters, the calculated strobetrack cross position does not necessarily have to be the geometricalmean value of the closest points on the two strobe tracks. One may alsoconsider the probability density functions as well as other aspects. Inthis calculation, a transformation is made from the respective localsensor coordinate systems, normally in spherical coordinates, to aglobal coordinate system, usually in Cartesian coordinates.

The preferred way to perform these calculations are as follows. With thenotation from above we have for 2D-strobe tracks (c.f. relations (12a)and (12b)):

P_(i)=(P_(θθ,i) cos²(φ_(i))e_(θ,i){circle around(X)}e_(θ,i)+P_(φφ,i)e_(φ,i){circle around (X)}e_(φ,i))d_(i) ²,  (17a)

P_(i) ⁻¹=(P_(θθ,i) ⁻¹ cos⁻²(φ)e_(θ,i){circle around (X)}e_(θ,i)+P_(θθ,i)⁻¹e_(φ,i){circle around (X)}e_(φ,i))d_(i) ⁻²,  (17b)

and for 1D-strobe tracks (c.f. relations (7a) and (7b)):

P_(i)=P_(θθ,i)e_(θ,i){circle around (X)}e_(θ,i)d_(i) ²,  (18a)

P_(i) ⁻¹=P_(θθ,i) ⁻¹e_(θ,i){circle around (X)}e_(θ,i)d_(i) ⁻²,  (18b)

for the two strobe tracks i=1,2. The estimated strobe track crossposition and its covariance is given by

X=(P₁ ⁻¹+P₂ ⁻¹)⁻¹(P₁ ⁻¹F₁+P₂ ⁻¹F₂),

P=(P₁ ⁻¹+P₂ ⁻¹)⁻¹.  (19)

When computing the estimated position X and its covariance P, theentries must be represented in the same coordinate system, preferablythe ET system.

If both strobe tracks are 1D, then we need to add a fictions observationat a default height and an uncertainty that covers all altitudes ofinterest. This is achieved by

 X=(P_(h) ⁻¹+P₁ ⁻¹+P₂ ⁻¹)⁻¹(P_(h) ⁻¹X₀+P₁ ⁻¹F₁+P₂ ⁻¹F₂),

P=(P_(h) ⁻¹+P₁ ⁻¹+P₂ ⁻¹)⁻¹.  (20)

Where

P_(h)=P_(RR)e_(R){circle around (X)}e_(R),

e_(R) is a unit vector pointing to the centre of the earth at theapproximate target position, and X₀ is the a priori height and P_(RR) isthe a priori height covariance.

From the above described calculations, the covariance matrix between thedifferent parameters as well as the probability density function of allparameters are acquired.

The next step in the recursive method of calculating strobe trackcrosses, uses the 2nd order strobe track crosses to calculate higherorder strobe track crosses. Strobe track crosses of order n is thuscalculated by using the information associated with strobe track crossesof order n−1, where n>2.

First, combinations of strobe track crosses of order n−1 are selected tocover all possible combinations of possible strobe track crosses oforder n. These pairs of (n−1)th order strobe track crosses are based onn−2 common original strobe tracks and the total number of used strobetracks are therefore n. A distance between two strobe track crosses oforder n−1 is calculated for performing a gating. The gating criteria canbe use any kind of distance measure, but preferably the statisticaldistance in a Cartesian global coordinate system is used. This preferreddistance measure is as follows. If the two 2-crosses are represented by(X₁,P₁) and (X₂,P₂), then the distance is the standard statisticaldistance:

 {square root over ((X₁+L -X₂+L )(P₁+L +P₂+L )⁻¹+L (X₁+L -X₂+L))}.  (21)

Note that the two distributions are not independent. However, for acoarse gating process it is operating satisfactorily.

The calculated minimum distance value is compared to a predeterminedthreshold value, and if the calculated minimum distance value exceedsthe threshold, the pair of strobe track crosses of order n−1 is veryunlikely as a candidate for a strobe track cross of order n and is thusrejected. The predetermined value could be a fixed value, a value chosenby the operator, a value depending on the uncertainties of the strobetrack states or a combination thereof.

The remaining pairs of strobe track crosses of order n−1, which havepassed the coarse gating processes described above, will give rise to anth order strobe track cross. The strobe track cross position and theassociated uncertainty is calculated. This calculation uses theinformation which is available from the strobe track crosses of order(n−1). These calculations may be performed directly from the originalstrobe track states, but since many of the desired calculations alreadyare performed during the calculation of the strobe track crosses oforder n−1, many results may be used directly. Since not only theestimated angles of the strobe track states are available, but alsoestimated angular velocities and covariance's associated with theseparameters, the calculated strobe track cross position does notnecessarily have to be the geometrical mean value of the closest pointson the two strobe tracks. One may also consider the probability densityfunctions as well as other aspects.

The preferred way to perform these calculations are as follows. We havefor 2D-strobe tracks (c.f. (12b)):

 P_(i) ⁻¹=(P_(θθ,i) ⁻¹ cos⁻²(φ)e_(θ,i){circle around(X)}e_(θ,i)+P_(φφ,i) ⁻¹e_(φ,i){circle around (X)}e_(φ,i))d_(i) ⁻²,  (22)

and for 1D-strobe tracks (c.f. (7b)):

P_(i) ⁻¹=(P_(θθ,i) ⁻¹e_(θ,i){circle around (X)}e_(θ,i))d_(i) ⁻²,  (23)

for the n strobe tracks, i=1, 2 . . . n. Then the estimated position andits covariance is given by $\begin{matrix}{{X = {( {\sum\limits_{i = 1}^{n}P_{i}^{- 1}} )^{- 1}( {\sum\limits_{i = 1}^{n}{P_{i}^{- 1}F_{i}}} )}},{P = {( {\sum\limits_{i = 1}^{n}P_{i}^{- 1}} )^{- 1}.}}} & (24)\end{matrix}$

If all strobe tracks are 1D, then it is required to add a fictionsobservation at a default height and an uncertainty that covers allaltitudes of interest as described above. The result is then$\begin{matrix}{{X = {( {P_{h}^{- 1} + {\sum\limits_{i = 1}^{n}P_{i}^{- 1}}} )^{- 1}( {{P_{h}^{- 1}X_{0}} + {\sum\limits_{i = 1}^{n}{P_{i}^{- 1}F_{i}}}} )}},{P = ( {P_{h}^{- 1} + {\sum\limits_{i = 1}^{n}P_{i}^{- 1}}} )^{- 1}},} & (25)\end{matrix}$

with the same P_(h) and X₀ as above. From the above describedcalculations, the covariance matrix between the different parameters, aswell as the probability density function of all parameters are acquiredfor the strobe track crosses of order n.

The next step in the initiation procedure is the step of selecting onestrobe track cross as a tentative target 13 (FIG. 2). According to thepresent invention there should be a process in which the most probablestrobe track cross is selected. In a preferred embodiment of theinvention, this selection is performed by defining a so called crossquality (XQ) value and to calculate this value for each strobe trackcross. FIG. 8 shows a block diagram illustrating the selecting step. Theprocess starts with the calculation of the XQ value for all strobe trackcrosses 31. In the next step, the strobe track crosses are sorted indescending XQ order 32 into a list of tentative targets. From this list,at least one strobe track cross is selected as a probable target. Thisstep is shown in FIG. 8 as step 33. Preferably the first strobe trackcross in the list, i.e. the strobe track cross with the highest XQ valueis selected. Once a strobe track has been used to create a target track,the strobe track is unlikely to participate in any target. It istherefore very likely that the strobe tracks contributing to theselected strobe track cross only participate in ghost crosses beside theselected strobe track cross. Strobe track crosses that havecontributions from at least one of the strobe tracks, which contributeto the selected strobe track cross, are therefore assumed to be ghostsand are subsequently eliminated from the list in step 34. If there isonly one target present, the elimination step 34 will empty the list ofstrobe track crosses. However, if there are more targets to beidentified, some strobe track crosses remain. Step 35 will examine ifthere are any strobe track crosses left in the list, and in such a casethe process returns to step 33 again to select another strobe trackcross for another probable target. If the list is empty, the processcontinues to step 36 which is a target track initiating step. This stepwill be discussed more in detail below.

The calculation should preferably consider the consistency of theparameters of the strobe track cross, especially if compared with otherstrobe track crosses using the same strobe tracks. The cross qualitycould also consider the order of the strobe track cross, since a higherorder strobe track cross in general is more probable to correspond to atrue target than a lower order strobe track cross. The parameters whichshould be considered comprise in general positions and velocities, butcould also comprise other characteristic parameters detectable by thesensors, such as the type of the target (if available) etc.

In the preferred embodiment of this invention, the cross quality valuecalculation is based on the probability density functions of the strobetracks contributing to the strobe track cross. Since most of thequantities for such calculations already have been calculated during thestrobe track cross position calculations, such part results may be usedand the XQ value calculation can be made comparatively fast. Thefollowing cross quality definition is preferred.

Given a strobe track cross X=X_(i) _(t) _(, i) ₂ _(. . . i) _(n) oforder n, formed from strobe tracks {ST_(k,i) _(k) }_(k=1 . . . n), thequality can be computed the as: $\begin{matrix}{{{X\quad {Q(X)}} = {\prod\limits_{k = {1\ldots \quad n}}\quad {X\quad {Q( {S\quad T_{k,i_{k}}} )}}}},} & (26)\end{matrix}$

where $\begin{matrix}{{X\quad {Q( {S\quad T_{k,i_{k}}} )}} = {\frac{{f_{k}(X)}{P(X)}}{{\sum\limits_{X^{\prime} \in {S\quad T_{k,i_{k}}}}{{f_{k}( X^{\prime} )}{P( X^{\prime} )}}} + {{\rho ( {S\quad T_{k,i_{k}}} )}{P( {S\quad T_{k,i_{k}}} )}}}.}} & (27)\end{matrix}$

The sum runs over all strobe track crosses that the strobe trackcontributes to and ƒ_(k)(X) is the distribution associated with thestrobe track, linearised and computed at the estimated location. Thedensity ρ gives the probability that a target is seen by one sensoronly, and is normally a constant.

The a priori probabilities P(X), P(ST) can be taken from differentsources. The information that tentative targets has been seen in aparticular region can be used to increase the value, or if it isunlikely to be any targets in a region the value is lowered. However,the main purpose is to increase the probability for strobe track crosseswith many strobe tracks, i.e. a strobe track cross of order n is morelikely than a strobe track cross of order n−1. This is an ad hoc method,which is due to approximations in design. For example, the XQ value fora 2^(nd) order strobe track cross is not decreased if it is not seen bya sensor which should be able to see it. A simple approach is to setP(X)=C^(n), where n is the order of the strobe track cross.

The initiating step 36 is described in more detail with reference toFIG. 9. One way to proceed is to let all the selected strobe trackcrosses result in initiation of a new target track. The selected strobetrack crosses are, however, associated with useful information, e.g. thecross quality value and the order of the strobe track cross. Thisinformation may be further used for controlling the further processingof the strobe track crosses. In many cases, a predetermined minimum XQvalue may exist, under which the strobe track cross is assumed to be toouncertain. Strobe track crosses which exhibit XQ values below thisthreshold could for instance just be shown as marks at the operatormonitor. In some cases, the operator might want to participate in theevaluation process of the strobe track crosses. Operator experiencecould be very useful in separating true targets from ghosts. Strobetrack crosses with very high XQ values or of a very high order mayautomatically be left for automatic target track initiation. Lower XQvalues or e.g. only 2nd order strobe track crosses might be presentedfor the operator to make an approval before initiation.

FIG. 9 shows a preferred embodiment of the initiation process. In step41, it is decided if the strobe track cross fulfils the criterion forautomatic target track initiation. If the criterion is not fulfilled,the next step 42 is to compare the information associated with thestrobe track cross to the criterion for allowing for a manual approvalfor target track initiation. Both of these criteria could involve the XQvalue, the order of the strobe track cross, or both. If the criterionfor a manual approval is fulfilled, the process waits for the operatorto decide on if the strobe track cross is approved for a target track ornot, represented by step 43. If the operator denies approval or if noneof the criteria are fulfilled, the strobe track cross is rejected as atarget candidate and is only shown as a mark at the operator monitor(step 44). If the strobe track cross is accepted as a candidate for anew target track, the process continues with the actual target trackinitiation, step 45, which is the same step as step 14 in FIG. 2.

The actual creation of the tracks is described below. The strobe trackscontributing to the selected strobe track cross contain informationabout angles and angular velocities of the strobe tracks as well astheir covariance's. From this amount of information a target trackvector is formed, containing e.g. a position, velocity and optionallyacceleration of the target. These parameters are preferably calculatedin a global tracking coordinate system (c.f. FIG. 14). Besides thetarget track vector itself, the strobe tracks may contribute in forminga covariance matrix to the target track vector. From this initial targettrack vector and its covariance matrix, a conventional trackingoperation can be started.

A preferred way to calculate the target track vector and its covariancematrix in a Cartesian coordinate system can be as follows. To initiate aKalman filter we need to specify the six dimensional state vector(x,{dot over (x)}) and its covariance $\quad \begin{pmatrix}P_{x\quad x} & P_{x\overset{.}{x}} \\P_{x\overset{.}{x}} & P_{\overset{.}{x}\overset{.}{x}}\end{pmatrix}$

(a 6×6 matrix). The estimators for the state vector is $\begin{matrix}{{x = {( {\sum\limits_{i = 1}^{n}P_{i}^{- 1}} )^{- 1}( {\sum\limits_{i = 1}^{n}{P_{i}^{- 1}F_{i}}} )}},} & (28) \\{{\overset{.}{x} = {( {\sum\limits_{i = 1}^{n}P_{i}^{- 1}} )^{- 1}( {\sum\limits_{i = 1}^{n}{P_{i}^{- 1}m_{i}}} )}},} & (29)\end{matrix}$

where

m_(i)=d_(i)({dot over (θ)}_(i) cos(φ_(i))e_(θ,i)+{dot over(φ)}_(i)e_(φ,i))  (30)

for a 2D-strobe track and

m_(i)=d_(i){dot over (θ)}_(i)e_(θ,i)  (31)

for a 1D-strobe track. The covariance is estimated as follows:$\begin{matrix}{{P_{x\quad x} = ( {\sum\limits_{k = 1}^{n}P_{k}^{- 1}} )^{- 1}},{P_{\overset{.}{x}\quad \overset{.}{x}} = {( {\sum\limits_{k = 1}^{n}P_{k}^{- 1}} )^{- 1}( {\sum\limits_{k = 1}^{n}{P_{k}^{- 1}V_{k}P_{k}^{- 1}}} )( {\sum\limits_{k = 1}^{n}P_{k}^{- 1}} )^{- 1}}},{P_{\overset{.}{x}\quad x} = {( {\sum\limits_{k = 1}^{n}P_{k}^{- 1}} )^{- 1}( {\sum\limits_{k = 1}^{n}{P_{k}^{- 1}M_{k}P_{k}^{- 1}}} )( {\sum\limits_{k = 1}^{n}P_{k}^{- 1}} )^{- 1}}},} & (32)\end{matrix}$

with

V≈(P_({dot over (θ)}{dot over (θ)}) cos²(φ)e_(θ){circle around(X)}e_(θ)+P_({dot over (φ)}{dot over (φ)})e_(φ){circle around(X)}e_(φ))d²,

M≈(P_(θ{dot over (θ)}) cos²(φ)e_(θ){circle around(X)}e_(θ)+P_(φ{dot over (φ)})e_(φ){circle around (X)}e_(φ))d²

for 2D-strobe tracks and

V≈P_({dot over (θ)}{dot over (θ)})e_(θ){circle around (X)}e_(θ)d²,

M≈P_(θ{dot over (θ)})e_(θ){circle around (X)}e_(θ)d²

for 1D-strobe tracks. The formulas above can be simplified using theapproximations${{P^{- 1}V\quad P^{- 1}} \approx {( {{\frac{P_{\overset{.}{\theta}\overset{.}{\theta}}}{P_{\theta\theta}^{2}{\cos^{2}(\phi)}}{e_{\theta} \otimes e_{\theta}}} + {\frac{P_{\overset{.}{\phi}\overset{.}{\phi}}}{P_{\phi\phi}^{2}}{e_{\phi} \otimes e_{\phi}}}} )d^{- 2}}},{{P^{- 1}M\quad P^{- 1}} \approx {( {{\frac{P_{\theta \overset{.}{\theta}}}{P_{\theta\theta}^{2}{\cos^{2}(\phi)}}{e_{\theta} \otimes e_{\theta}}} + {\frac{P_{\phi \overset{.}{\phi}}}{P_{\phi\phi}^{2}}{e_{\phi} \otimes e_{\phi}}}} )d^{- 2}}}$

for 2D-strobe tracks and${{P^{- 1}V\quad P^{- 1}} \approx {\frac{P_{\overset{.}{\theta}\overset{.}{\theta}}}{P_{\theta\theta}^{2}}{e_{\theta} \otimes e_{\theta}}d^{- 2}}},{{P^{- 1}M\quad P^{- 1}} \approx {\frac{P_{\theta \overset{.}{\theta}}}{P_{\theta\theta}^{2}}{e_{\theta} \otimes e_{\theta}}d^{- 2}}}$

for 1D-strobe tracks. If all strobe tracks above are 1D then we alsoneed to add the stabilising factor P_(h) to position the strobe trackcross at an a priori height.

Once a target track is established, the maintenance is quite similar toprior art. The target tracks are updated by suitable individual sensorstrobes, which fall within a gate around the target. The sensor strobeswhich are used in this way are thereafter eliminated and do thereforenot participate in maintaining the strobe tracks. This means that when atarget track is established it will consume relevant sensor strobes andcause the corresponding strobe tracks to starve and subsequently bedeleted. In connection with this, the amount of calculation will bereduced.

When a target disappears, or leaves the area of interest, the targettrack should also be deleted. As in previous techniques, this will bethe case when there are not enough new strobes to keep the target trackuncertainties below a certain level, or if a certain time period haselapsed since the last useful strobe.

Some details of the initiation process will be described in detail bymeans of a few explanatory examples.

EXAMPLE 1

This example is described with references to FIG. 10. In this example,the tracking system consists of two 2D sensors D1 and D2, working intheir local coordinate systems C1 and C2, respectively. One single truetarget T is present within the range of the sensors. Sensor D1 obtains aseries of strobes associated with the target T and a strobe track ST11is initiated. The strobe track state yields estimates of azimuth angle,angular velocity, elevation angle and angular and a covariance, c.f.(2): $\begin{pmatrix}\theta_{1} \\{\overset{.}{\theta}}_{1}\end{pmatrix}\quad,\quad \begin{pmatrix}P_{{\theta\theta},1} & P_{{\theta \overset{.}{\theta}},1} \\P_{{\overset{.}{\theta}\theta},1} & P_{{\overset{.}{\theta}\overset{.}{\theta}},1}\end{pmatrix}$ $\begin{pmatrix}\phi_{1} \\{\overset{.}{\phi}}_{1}\end{pmatrix}\quad,\quad {\begin{pmatrix}P_{{\phi\phi},1} & P_{{\phi \overset{.}{\phi}},1} \\P_{{\overset{.}{\phi}\phi},1} & P_{{\overset{.}{\phi}\overset{.}{\phi}},1}\end{pmatrix}.}$

In a similar way, sensor D2 initiates a strobe track ST21, withcorresponding estimates but with index 1 replaced by 2. Next, therepresentation of the strobe track is computed in the sensor system (SS)according to (9):

e_(r,1)=(sin(θ₁)cos(φ₁), cos(θ₁)cos(φ₁), sin(φ₁)),

e_(θ,1)=(cos(θ₁)cos(φ₁), −sin(θ₁)cos(φ₁), sin(φ₁)),

e_(φ,1)=(−sin(θ₁)sin(φ₁), −cos(θ₁)sin(φ₁), cos(φ₁)).

The strobe track can now be parametrised according to (10) as follows:

 l₁(d)=e_(r,i)d+F_(i), d>0.

Corresponding computation is carried out for the other strobe track,where similar result is obtained, but with index 1 replaced with 2.Next, the representations of the strobe tracks are transformed to the ETsystem (1). The points of linearisation can now be computed (15) as:$( {d_{1},d_{2}} ) = {\underset{d_{1},{d_{2} > 0}}{argmin}( {{e_{r,1}d_{1}} + F_{1} - {e_{r,2}d_{2}} - F_{2}} )}^{2}$

The linerised normal distribution (11)-(13) for strobe track 1 is:

N(e_(r,1)d₁+F₁,P₁),

P₁=(P_(θθ,1) cos²(φ₁)e_(θ,1){circle around(X)}e_(θ,1)+P_(φφ,1)e_(φ,1){circle around (X)}e_(φ,1))d₁ ².

P₁ ⁻¹=(P_(θθ,1) ⁻¹ cos⁻²(φ₁e_(θ,1){circle around (X)}e_(θ,1)+P_(φφ,1)⁻¹e_(φ,1){circle around (X)}e_(φ,1))d₁ ⁻²,

${f_{1}(x)} = {\frac{1}{2\pi \sqrt{P_{{\theta\theta},1}P_{{\phi\phi},1}}}{{\exp ( {{- ( {x - F_{1}} )}{{P^{- 1}( {x - F_{1}} )}/2}} )}.}}$

The corresponding computation is carried out for strobe track 2. Ifd₁,d₂ do not exceed the maximum range of the sensor, the statisticaldistance may then be computed according to (16):$\frac{( {F_{1} - F_{2}} ) \cdot e_{0}}{\sqrt{\begin{matrix}{{( {{P_{{\theta\theta},1}( {e_{0} \cdot e_{\theta,1}} )}^{2} + {P_{{\phi\phi},1}( {e_{0} \cdot e_{\phi,1}} )}^{2}} )d_{1}^{2}} +} \\{( {{P_{{\theta\theta},2}( {e_{0} \cdot e_{\theta,2}} )}^{2} + {P_{{\phi\phi},2}( {e_{0} \cdot e_{\phi,2}} )}^{2}} )d_{2}^{2}}\end{matrix}}},{e_{0} = {e_{r,1} \times {e_{r,2}.}}}$

If the statistical distance is smaller than a given a priori gate thenthe strobe track cross is accepted, otherwise it is rejected. In thisexample it is assumed to be accepted and the next step is to estimatethe position of the strobe track cross (19):

 X₁₁=(P₁ ⁻¹+P₂ ⁻¹)⁻¹(P₁ ⁻¹F₁+P₂ ⁻¹F₂),

P₁₁=(P₁ ⁻¹+P₂ ⁻¹)⁻¹.

Since all the entries are computed in the ET-system, the strobe trackcross and its covariance is obtained in the same system.

Next, the quality of the strobe track cross is computed, c.f. (26),(27):

XQ(X₁₁)=XQ(ST₁₁)XQ(ST₂₁),

where${{X\quad {Q( {S\quad T_{11}} )}} = \frac{{f_{1}( X_{11} )}{P( X_{11} )}}{{{f_{1}( X_{11} )}{P( X_{11} )}} + {{\rho ( {S\quad T_{11}} )}{P( {S\quad T_{11}} )}}}},{{X\quad {Q( {S\quad T_{21}} )}} = {\frac{{f_{2}( X_{11} )}{P( X_{11} )}}{{{f_{2}( X_{11} )}{P( X_{11} )}} + {{\rho ( {S\quad T_{21}} )}{P( {S\quad T_{21}} )}}}.}}$

Obviously strobe track cross X11 is the strobe track cross with highestquality, since it is the only one existing, and it will subsequently beselected. If the strobe track cross is to be used to initiate a targettrack, the Kalman filter state is initiated as follows, c.f. (28)-(32):

State vector=(x,{dot over (x)}) ${Covariance} = \begin{pmatrix}P_{x\quad x} & P_{x\overset{.}{x}} \\P_{x\overset{.}{x}} & P_{\overset{.}{x}\overset{.}{x}}\end{pmatrix}$

 x=(P₁ ⁻¹+P₂ ⁻¹)⁻¹(P₁ ⁻¹F₁+P₂ ⁻¹F₂),

{dot over (x)}=(P₁ ⁻¹+P₂ ⁻¹)⁻¹(P₁ ⁻¹m₁+P₂ ⁻¹m₂),

where

m₁=d₁({dot over (θ)}₁ cos(φ₁)e_(θ,1)+{dot over (φ)}₁e_(φ,1)),

m₂=d₂({dot over (θ)}₂ cos(φ₂)e_(θ,2)+{dot over (φ)}₂e_(φ,2)).

and

P_(xx)=(P₁ ⁻¹+P₂ ⁻¹)⁻¹,

P_({dot over (x)}{dot over (x)})=(P₁ ⁻¹+P₂ ⁻¹)⁻¹(P₁ ⁻¹V₁P₁ ⁻¹+P₂ ⁻¹V₂P₂⁻¹)(P₁ ⁻¹+P₂ ⁻¹)⁻¹,

P_({dot over (x)}x)=(P₁ ⁻¹+P₂ ⁻¹)⁻¹(P₁ ⁻¹M₁P₁ ⁻¹+P₂ ⁻¹M₂P₂ ⁻¹)(P₁ ⁻¹+P₂⁻¹)⁻¹,

where${P_{1}^{- 1}V_{1}\quad P_{1}^{- 1}} \approx {( {{\frac{P_{{\overset{.}{\theta}\overset{.}{\theta}},1}}{P_{{\theta\theta},1}^{2}{\cos^{2}( \phi_{1} )}}{e_{\theta,1} \otimes e_{\theta,1}}} + {\frac{P_{{\overset{.}{\phi}\overset{.}{\phi}},1}}{P_{{\phi\phi},1}^{2}}{e_{\phi,1} \otimes e_{\phi,1}}}} )d_{1}^{- 2}}$${P_{2}^{- 1}V_{2}\quad P_{2}^{- 1}} \approx {( {{\frac{P_{{\overset{.}{\theta}\overset{.}{\theta}},2}}{P_{{\theta\theta},2}^{2}{\cos^{2}( \phi_{2} )}}{e_{\theta,2} \otimes e_{\theta,2}}} + {\frac{P_{{\overset{.}{\phi}\overset{.}{\phi}},2}}{P_{{\phi\phi},2}^{2}}{e_{\phi,2} \otimes e_{\phi,2}}}} )d_{2}^{- 2}}$${P_{1}^{- 1}M_{1}\quad P_{1}^{- 1}} \approx {( {{\frac{P_{{\theta \overset{.}{\theta}},1}}{P_{{\theta\theta},1}^{2}{\cos^{2}( \phi_{1} )}}{e_{\theta,1} \otimes e_{\theta,1}}} + {\frac{P_{{\phi \overset{.}{\phi}},1}}{P_{{\phi\phi},1}^{2}}{e_{\phi,1} \otimes e_{\phi,1}}}} )d_{1}^{- 2}}$${P_{2}^{- 1}M_{2}\quad P_{2}^{- 1}} \approx {( {{\frac{P_{{\theta \overset{.}{\theta}},2}}{P_{{\theta\theta},2}^{2}{\cos^{2}( \phi_{2} )}}{e_{\theta,2} \otimes e_{\theta,2}}} + {\frac{P_{{\phi \overset{.}{\phi}},2}}{P_{{\phi\phi},2}^{2}}{e_{\phi,2} \otimes e_{\phi,2}}}} ){d_{2}^{- 2}.}}$

EXAMPLE 2

In this example, three 2D sensors are used and one true target ispresent within the range of the sensors, as seen in FIG. 11. The strobetrack crosses (X₁₁₀,P₁₁₀), (X₁₀₁,P₁₀₁) and (X₀₁₁,P₀₁₁), are computed asin example 1 above. To compute the 3rd order strobe track cross, the 2ndorder strobe track cross between sensor 1&2 and 1&3 are compared, asdescribed in FIG. 5 and (21) and the 3rd order strobe track cross(X₁₁₁,P₁₁₁) is accepted if and only if

 {square root over ((X₁₁₀+L −X₁₀₁+L )(P₁₁₀+L +P₁₀₁+L )⁻¹+L (X₁₁₀+L−X₁₀₁+L ))}<Predefined gate.

The new estimate of the 3rd order strobe track cross is (c.f. (24)):

X₁₁₁=(P₁ ⁻¹+P₂ ⁻¹+P₃ ⁻¹)⁻¹(P₁ ⁻¹F₁+P₂ ⁻¹F₂+P₃ ⁻¹F₃),

P₁₁₁=(P₁ ⁻¹+P₂ ⁻¹P₃ ⁻¹)⁻¹.

The cross quality (26) for a 2nd order strobe track cross is now

XQ(X₁₁₀)=XQ(ST₁₁)XQ(ST₂₁),

where (c.f. (27))${{X\quad {Q( {S\quad T_{11}} )}} = \frac{{f_{1}( X_{110} )}{P( X_{110} )}}{\begin{matrix}{{{f_{1}( X_{110} )}{P( X_{110} )}} + {{f_{1}( X_{101} )}{P( X_{101} )}} +} \\{{{f_{1}( X_{111} )}{P( X_{111} )}} + {{\rho ( {S\quad T_{11}} )}{P( {S\quad T_{11}} )}}}\end{matrix}}},{{X\quad {Q( {S\quad T_{21}} )}} = {\frac{{f_{2}( X_{110} )}{P( X_{110} )}}{\begin{matrix}{{{f_{2}( X_{110} )}{P( X_{110} )}} + {{f_{2}( X_{011} )}{P( X_{011} )}} +} \\{{{f_{2}( X_{111} )}{P( X_{111} )}} + {{\rho ( {S\quad T_{21}} )}{P( {S\quad T_{21}} )}}}\end{matrix}}.}}$

For the 3rd order strobe track cross, the quality (26) is obtained by:

XQ(X₁₁₁)=XQ(ST₁₁)XQ(ST₂₁)XQ(ST₃₁),

where (c.f. (27))${{X\quad {Q( {S\quad T_{11}} )}} = \frac{{f_{1}( X_{111} )}{P( X_{111} )}}{\begin{matrix}{{{f_{1}( X_{110} )}{P( X_{110} )}} + {{f_{1}( X_{101} )}{P( X_{101} )}} +} \\{{{f_{1}( X_{111} )}{P( X_{111} )}} + {{\rho ( {S\quad T_{11}} )}{P( {S\quad T_{11}} )}}}\end{matrix}}},{{X\quad {Q( {S\quad T_{21}} )}} = \frac{{f_{2}( X_{111} )}{P( X_{111} )}}{\begin{matrix}{{{f_{2}( X_{110} )}{P( X_{110} )}} + {{f_{2}( X_{011} )}{P( X_{011} )}} +} \\{{{f_{2}( X_{111} )}{P( X_{111} )}} + {{\rho ( {S\quad T_{21}} )}{P( {S\quad T_{21}} )}}}\end{matrix}}},{{X\quad {Q( {S\quad T_{31}} )}} = {\frac{{f_{3}( X_{111} )}{P( X_{111} )}}{\begin{matrix}{{{f_{2}( X_{101} )}{P( X_{101} )}} + {{f_{2}( X_{011} )}{P( X_{011} )}} +} \\{{{f_{2}( X_{111} )}{P( X_{111} )}} + {{\rho ( {S\quad T_{31}} )}{P( {S\quad T_{31}} )}}}\end{matrix}}.}}$

In this example it is assumed that XQ(X₁₁₁)>XQ(X₁₁₀)>XQ(X₀₁₁)>XQ(X₁₀₁).Accordingly, X₁₁₁ is selected as the first strobe track cross. Thestrobe tracks ST₁₁,ST₂₁,ST₃₁, have all at least one strobe track incommon with the selected strobe track cross and hence the strobe trackcrosses X₁₁₀,X₁₀₁,X₀₁₁ will be removed from the list. The list will inthis example now be empty and no more strobe track crosses are selected,which is in accordance with the initial model. The result from theentire process is that the strobe track cross (X₁₁₁,P₁₁₁) represents thepossible target position. A target track may now be initiated accordingto (28)-(32), compare also with example 1 above.

EXAMPLE 3

In this example, a case of three 2D-sensors and two true targets isconsidered, as shown in FIG. 12. The following strobe track crosses andtheir qualities are computed as in example 2 above: (X₁₁₀,P₁₁₀),(X₁₀₁,P₁₀₁), (X₀₁₁,P₀₁₁), (X₁₁₁,P₁₁₁), (X₂₁₀,P₂₁₀), (X₂₀₁,P₂₀₁),(X₀₂₁,P₀₂₁), (X₂₂₁,P₂₂₁), (X₂₂₀,P₂₂₀), (X₂₀₂,P_(202), (X)₀₂₂,P_(022), (X) ₂₂₂,P₂₂₂). The quality calculation yields the followingrelations

XQ(X₂₂₂)>XQ(X₂₂₀)>XQ(X₁₁₁)> . . .

In accordance with the above described selection procedure, strobe trackcross (X₂₂₂,P₂₂₂) is selected to initiate a target track in accordancewith what is described in example 1 above. Subsequently, the strobetracks ST₁₂,ST₂₂,ST₃₂ are considered to be consumed and hence thefollowing strobe track crosses are removed from the list: (X₂₁₀,P₂₁₀),(X₂₀₁,P₂₀₁), (X₀₂₁P₀₂₁), (X₂₂₁,P₂₂₁), (X₂₂₀,P₂₂₀), (X₂₀₂,P₂₀₂),(X₀₂₂,P₀₂₂). The next strobe track cross in the list to be selected toinitiate a target track is (X₁₁₁,P₁₁₁). Consequently, the strobe tracksST₁₁,ST₂₁,ST₃₁ are consumed and hence the following strobe track crossesare removed (X₁₁₀,P₁₁₀), (X₁₀₁,P₁₀₁), (X₀₁₁,P₀₁₁). The list ofhypothetical target crosses is now empty and the result of the totalanalysis is the creation of two new target tracks based on the strobetrack crosses (X₂₂₂,P₂₂₂) and (X₁₁₁,P₁₁₁).

From this example, the advantages of the present invention compared tothe prior art is also obvious. In FIG. 13, a part of FIG. 12 isenlarged, and for to explain the details, individual strobes, associatedwith the strobe tracks are shown as thin lines. As in FIG. 4,probability density functions are shown for the strobe tracks andcorresponding uncertainty distributions for the individual strobes. Inthis example, sensor D3 has a lower accuracy in the measurement, and thedistance between the probable target and the sensor is larger for sensorD3 than for the sensors D1 and D2. This results in that the PDF for theST31 strobe track is wider than for the other participating strobetracks ST11 and ST21. In a prior art evaluation of the 3rd order crossX111, the position of the cross would be put in the centre of gravity ofthe triangle built by some strobes. It is obvious that any positionderived in such a way will contain a large amount of uncertainty. Sincethe measurements by sensors D1 and D2 are more accurate, a weightedcross position should be used. Furthermore, by using strobe tracksinstead of individual strobes, a large amount of noise and uncertaintywill be filtered even before creating a target track, thus giving targettrack initiations which are much more reliable.

By using angular velocity information from the strobe tracks, clearinconsistencies may be discovered. In the above case, the X111 strobetrack cross corresponds to a true target, and thus the individualangular velocities V11, V21, V31 of the three strobe tracks areconsistent with each other. The V11 and V21 angular velocities areaccurate, and implies that the X111 velocity should be directed to theleft in the figure, which also is confirmed by the V31 velocity. If,however, the V31 would be directed in the opposite direction and theassociated uncertainty is too small to explain the divergence, the X111strobe track cross would be possible to reject as a ghost.

EXAMPLE 4

In this example, two 1D-sensors and one target is considered, as shownin FIG. 6. Since the algorithm is performed in three dimensions andheight information is completely missing, the problem in this example issingular. Therefore, it is necessary to add a priori height information.The unique strobe track cross is (X₁₁,P₁₁). The estimated position andits covariance is obtained from (20):

X₁₁=(P_(h) ⁻¹+P₁ ⁻¹+P₂ ⁻¹)⁻¹(P_(h) ⁻¹X₀+P₁ ⁻¹F₁+P₂ ⁻¹F₂),

P₁₁=(P_(h) ⁻¹+P₁ ⁻¹+P₂ ⁻¹)⁻¹

Here P₁,P₂,F₁,F₂ are the same as in example 1 above. The new termsX₀,P_(h) are needed to make P₁ ⁻¹+P₂ ⁻¹ invertible.

The computation is performed as follows. The intersection of the twoplanes representing strobe track one and two is a line which can beparametrised as

t→X₀+e_(R)t,

where X₀ is a point on the line and e_(R) is a unit vector parallel tothe line. X₀ is selected so that its height over mean sea level (msl) isthat of a predefined value. In this example 7000 meters is chosen. Thecovariance is then

 P_(h)=P_(RR)e_(R){circle around (X)}e_(R),

where the scalar P_(RR) is selected in such a way that the targets ofinterest are in the height range [7000−{square root over (P_(RR)+L )},7000+{square root over (P_(RR)+L )}]. In this example P_(RR)=7000*7000.The quality of the strobe track cross can not be computed, however, itis unique and hence selected as a possible target position.

The initial Kalman filter state is (x,{dot over (x)}), $\begin{pmatrix}P_{x\quad x} & P_{x\overset{.}{x}} \\P_{x\overset{.}{x}} & P_{\overset{.}{x}\overset{.}{x}}\end{pmatrix}$

where, according to (28), (29), (31) and (32)

x=(P_(h) ⁻¹+P₁ ⁻¹+P₂ ⁻¹)⁻¹(P_(h) ⁻¹X₀+P₁ ⁻¹F₁+P₂ ⁻¹F₂),

{dot over (x)}=(P_(h) ⁻¹+P₁ ⁻¹+P₂ ⁻¹)⁻¹(P₁ ⁻¹m₁+P₂ ^(−·)m₂),

where

m₁ =d₁ θ₁ e_(θ,1)

m₂ =d₂ θ₂ e_(θ,2)

P_(xx)=(P_(h) ⁻¹+P₁ ⁻¹+P₂ ⁻¹)⁻¹,

P_({dot over (x)}{dot over (x)})=(P_(h) ⁻¹+P₁ ⁻¹+P₂ ⁻¹)⁻¹(P₁ ⁻¹V₁P₁⁻¹+P₂ ⁻¹V₂P₂ ⁻¹)(P_(h) ⁻¹+P₁ ⁻¹+P₂ ⁻¹)⁻¹,

P_(x{dot over (x)})=(P_(h) ⁻¹+P₁ ⁻¹+P₂ ⁻¹)⁻¹(P₁ ⁻¹M₁P₁ ⁻¹+P₂ ⁻¹M₂P₂⁻¹)(P_(h) ⁻¹+P₁ ⁻¹+P₂ ⁻¹)⁻¹,

with${{P_{1}^{- 1}V_{1}P_{1}^{- 1}} \approx {\frac{P_{{\overset{.}{\theta}\overset{.}{\theta}},1}}{P_{{\theta\theta},1}^{2}}{e_{\theta,1} \otimes e_{\theta,1}}d_{1}^{- 2}}},{{P_{2}^{- 1}V_{2}P_{2}^{- 1}} \approx {\frac{P_{{\overset{.}{\theta}\overset{.}{\theta}},2}}{P_{{\theta\theta},2}^{2}}{e_{\theta,2} \otimes e_{\theta,2}}d_{2}^{- 2}}},{{P_{1}^{- 1}M_{1}P_{1}^{- 1}} \approx {\frac{P_{{\theta \overset{.}{\theta}},1}}{P_{{\theta\theta},1}^{2}}{e_{\theta,1} \otimes e_{\theta,1}}d_{1}^{- 2}}},{{P_{2}^{- 1}M_{2}P_{2}^{- 1}} \approx {\frac{P_{{\theta \overset{.}{\theta}},2}}{P_{{\theta\theta},2}^{2}}{e_{\theta,2} \otimes e_{\theta,2}}{d_{2}^{- 2}.}}}$

The above described method is discussed assuming the absence ofatmospheric refraction. It is straightforward to someone skilled in theart to make the compensation when constructing hypothetical crosses,computing the XQ values, and in tracking association and updating. Auseful default compensation is where the height above mean sea level ofthe target is computed with the radius of the earth changed to kR, witha typical value of k=4/3. An approximate correction of the measuredelevation follows the formula:${- \frac{d}{2}}{R \cdot \frac{k - 1}{k}}$

The value k=4/3 is dependent on the weather, type of sensors and manyother parameters and may be changed readily. These compensations are,however, already known, for instance by “Introduction to Radar Systems”by Merrill I. Skolnik, McGraw-Hill Book Company, 1981, pages 447-450.

What is claimed is:
 1. A track initiation method for multi targettracking by means of at least two passive sensors, comprising the stepsof: creating strobe tracks for each of the at least two sensors, wheresaid strobe tracks are filtered sets of strobes, belonging to the sametarget; calculating strobe track crosses, including a step ofcalculating 2nd order strobe track crosses; selecting a strobe trackcross as a probable target; and creating a target track, wherein saidstep of calculating 2nd order strobe track crosses comprises, for everypair of two said strobe tracks from different said sensors, the stepsof: calculating a distance between said strobe tracks; rejecting thestrobe track combination as a potential strobe track cross if thedistance is larger than a predetermined value; calculating a closestpoint on each strobe track corresponding to said distance between saidstrobe tracks; calculating a cross-to-sensor distance between eachclosest point and each respective sensor; rejecting the strobe trackcombination as a potential strobe track cross if at least one of thecross-to-sensor distances exceeds a range of a corresponding saidsensor; and calculating position and position uncertainty for strobetrack crosses of the remaining strobe tracks combinations.
 2. A trackinitiation method according to claim 1, wherein the step of calculatingstrobe tracks crosses, following said step of calculating 2nd orderstrobe track crosses, further comprises a step of calculating strobetrack crosses of order n, wherein n>2, and wherein each said calculationof strobe track crosses of order n>2 is based on strobe track crosses oforder n−1.
 3. A track initiation method according to claim 2, whereinsaid step of calculating strobe tracks crosses of order n, where n>2,further comprises steps of: calculating a distance between a combinationof two strobe track crosses of order n−1; rejecting a potential nthorder strobe track cross if the distance between said combination of twostrobe track crosses of order n−1 is larger than a predetermined value;and calculating position and position uncertainty for strobe trackcrosses of remaining combinations of strobe track crosses of order n−1.4. A track initiation method according to claim 1, wherein, if at leastone strobe track is a 2D strobe track, said step of calculating positionand position uncertainty is performed according to${X = {( {\sum\limits_{i = 1}^{n}P_{i}^{- 1}} )^{- 1}( {\sum\limits_{i = 1}^{n}{P_{i}^{- 1}F_{i}}} )}},{P = ( {\sum\limits_{i = 1}^{n}P_{i}^{- 1}} )^{- 1}}$

where P_(i) ⁻¹=(P_(θθ,i) ⁻¹ cos⁻²(φ)e_(θ,i){circle around(X)}e_(θ,i)+P_(φφ,i) ⁻¹e_(φ,i){circle around (X)}e_(φ,i))d_(i) ⁻², for a2D strobe track, and P_(i) ⁻¹=(P_(θθ,i) ⁻¹e_(θ,i){circle around(X)}e_(θ,i))d_(i) ⁻² for a 1D strobe track, P_(θθ,i) and P_(φφ,i)representing the state vector covariances of the strobe tracks.
 5. Atrack initiation method according to claim 1, wherein said distancebetween the strobe tracks (l₁,l₂) is computed according to${{Dist}( {l_{1},l_{2}} )} = \frac{\Delta}{\sqrt{P}}$

where Δ=(F₁−F₂)·e₀, P=(P_(θθ,1)(e₀·e_(θ,1))²+P_(φφ,1)(e₀·e_(φ,1))²)d₁²+(P_(θθ,2)(e₀·e_(θ,2))²+P_(φφ,2)(e₀·e_(φ,2))²)d₂ ²l_(i)(d)=e_(r,i)d+F_(i), d>0, i=1,2, (d₁,d₂)=arg_(d) ₁ _(,d) ₂ _(>0) min(e_(r,1)d₁+F₁−e_(r,2)d₂−F₂)², F_(i) representing the foot point vectorsof the respective sensors.
 6. A track initiation method according toclaim 3, wherein said distance between strobe track crosses is computedaccording to dist((X₁,P₁),(X₂,P₂))={square root over ((X₁+L −X₂+L )(P₁+L+P₂+L )⁻¹+L (X₁+L −X₂+L ))}. where the two strobe track crosses of 2ndorder are represented by (X₁,P₁) and (X₂,P₂), respectively.
 7. A trackinitiation method according to claim 1, wherein said step of selecting astrobe track cross as a probable target comprises the steps of:calculating a cross quality value for each strobe track cross using theprobability density functions of the strobe tracks contributing to thestrobe track cross; sorting strobe track crosses into a list in order bythe cross quality value; and selecting at least one said strobe trackcross from the list.
 8. A track initiation method according to claim 7,wherein said cross quality value calculation uses the order of thestrobe track cross.
 9. A track initiation method according to claim 8,wherein said cross quality value is calculated by${{X\quad {Q(X)}} = {\prod\limits_{k = {1\ldots \quad n}}\quad {X\quad {Q( {S\quad T_{k,i_{k}}} )}}}},$

where${{X\quad {Q( {S\quad T_{k,i_{k}}} )}} = \frac{{f_{k}(X)}{P(X)}}{{\sum\limits_{X^{\prime} \in {S\quad T_{k,i_{k}}}}{{f_{k}( X^{\prime} )}{P( X^{\prime} )}}} + {{\rho ( {S\quad T_{k,i_{k}}} )}{P( {S\quad T_{k,i_{k}}} )}}}},$

X=X_(i) ₁ _(,i) ₂ _(. . . i) _(n) a strobe track cross of order n,formed from strobe tracks {ST_(k,i) _(k) }_(k=1 . . . n).
 10. A trackinitiation method according to claim 7, wherein said step of selecting astrobe track cross as a probable target further comprises the steps of:removing those strobe track crosses, which are formed by at least onestrobe track that is used for forming the selected strobe track cross,from the sorted list; and repeating the selection of strobe trackcrosses as probable targets until the sorted list is empty.
 11. A trackinitiation method according to claim 7, wherein said step of selecting astrobe track cross as a probable target further comprises the step ofdetermining if a selected strobe track cross should give rise to anautomatic creation of a new target track, based on at least one of thecross quality value and the order of the strobe track cross.
 12. A trackinitiation method according to claim 1, wherein said step of creating atarget track comprises the step of calculating a target track statevector and its covariance matrix in a global tracking coordinate system.13. A track initiation method according to claim 12, wherein said stepof calculating a target track state vector and its covariance matrix isperformed in a Cartesian coordinate system according to$( {x,\overset{.}{x}} ),\quad \begin{pmatrix}P_{x\quad x} & P_{x\overset{.}{x}} \\P_{x\overset{.}{x}} & P_{\overset{.}{x}\overset{.}{x}}\end{pmatrix},$

where${x = {( {\sum\limits_{i = 1}^{n}P_{i}^{- 1}} )^{- 1}( {\sum\limits_{i = 1}^{n}{P_{i}^{- 1}F_{i}}} )}},{\overset{.}{x} = {( {\sum\limits_{i = 1}^{n}P_{i}^{- 1}} )^{- 1}( {\sum\limits_{i = 1}^{n}{P_{i}^{- 1}m_{i}}} )}},{m_{i} = {d_{i}( {{{\overset{.}{\theta}}_{i}{\cos ( \phi_{i} )}e_{\theta,i}} + {{\overset{.}{\phi}}_{i}e_{\phi,i}}} )}},{P_{x\quad x} = ( {\sum\limits_{k = 1}^{n}P_{k}^{- 1}} )^{- 1}},{P_{\overset{.}{x}\quad \overset{.}{x}} = {( {\sum\limits_{k = 1}^{n}P_{k}^{- 1}} )^{- 1}( {\sum\limits_{k = 1}^{n}{P_{k}^{- 1}V_{k}P_{k}^{- 1}}} )( {\sum\limits_{k = 1}^{n}P_{k}^{- 1}} )^{- 1}}},{P_{x\quad \overset{.}{x}} = {( {\sum\limits_{k = 1}^{n}P_{k}^{- 1}} )^{- 1}( {\sum\limits_{k = 1}^{n}{P_{k}^{- 1}M_{k}P_{k}^{- 1}}} )( {\sum\limits_{k = 1}^{n}P_{k}^{- 1}} )^{- 1}}},$

with${{P^{- 1}V\quad P^{- 1}} \approx {( {{\frac{P_{\overset{.}{\theta}\overset{.}{\theta}}}{P_{\theta\theta}^{2}{\cos^{2}(\phi)}}{e_{\theta} \otimes e_{\theta}}} + {\frac{P_{\overset{.}{\phi}\overset{.}{\phi}}}{P_{\phi\phi}^{2}}{e_{\phi} \otimes e_{\phi}}}} )d^{- 2}}},{{P^{- 1}M\quad P^{- 1}} \approx {( {{\frac{P_{\theta \overset{.}{\theta}}}{P_{\theta\theta}^{2}{\cos^{2}(\phi)}}{e_{\theta} \otimes e_{\theta}}} + {\frac{P_{\phi \overset{.}{\phi}}}{P_{\phi\phi}^{2}}{e_{\phi} \otimes e_{\phi}}}} ){d^{- 2}.}}}$